Macro-Finance ATSM with Bayesian

Main References

  • Joslin, S., Priebsch, M., & Singleton, K. J. (2014). Risk premiums in dynamic term structure models with unspanned macro risks. The Journal of Finance, 69(3), 1197-1233.
  • Kim, Seung Hyun and Kang, Kyu Ho, When Falling Starts hit the Zero Lower Bound: Implications for Dynamic Term Premiums (June 30, 2024). Available at SSRN: https://ssrn.com/abstract=4716351 or http://dx.doi.org/10.2139/ssrn.4716351
  • 이선호. (2023). 한국 수익률곡선에 대한 비생성 거시 위험의 국면 전환적 존재 여부 분석. 한국은행 경제연구원 ｢經濟分析｣ 제29권 제3호 (2023.9)
  • Lee, Sunho and Kang, Kyu H., Regime-Switching Macro Risks in the Term Structure of Interest Rates (August 9, 2023). Available at SSRN: https://ssrn.com/abstract=4414404 or http://dx.doi.org/10.2139/ssrn.4414404
  • Chib, S., & Kang, K. H. (2017). Efficient posterior sampling in Gaussian affine term structure models.
  • Lee, Sunho and Kang, Kyu H., A Bayesian Large Vector Autoregression of the Yield Curve and Macroeconomic Variables with No-Arbitrage Restriction (May 20, 2024). Available at SSRN: https://ssrn.com/abstract=4708628 or http://dx.doi.org/10.2139/ssrn.4708628
  • Kim, Seung Hyun. (2024). "Asset Pricing and Term Structure Models". WORK IN PROGRESS.

Unspanned Macro-Finance Model

Macro-Finance Models

Following Ang and Piazzesi (2003),

Unspanned Model Restrictions

Stylized Facts

Following Joslin, Priebsch, and Singleton (2014),

  • SF1: small number of risk factors, . thus and . Thus and the restriction is ensued by letting since and . Thus we have
  • SF2: macro risks unspanned by the yield curve, i.e. . and thus we need to ensure that , and also by letting , we have
  • SF3: macro factors help predict excess bond returns, i.e.

Arbitrage-free Nelson Siegel Restrictions

We use Identification of ATSMs > Arbitrage-free Nelson-Siegel Model from Niu and Zeng (2012):

  • unconditional mean of short rate is fixed: and we will denote it as . Since the short rate has presented long-run downward trend and high persistence, making the estimation inefficient.
  • restrictions on risk-neutral dynamics:
  • identification restrictions: to identify the latent factor, we additionally assume and

Then we have where and the initial condition is and .

Also,

Model Equations

We summarize up the whole model equations under the above restrictions:

  • short rate dynamics:

  • risk-neutral dynamics: and the full equation is where .

  • physical dynamics:

  • market prices of risk: where

  • stochastic discount factor: where .

  • Ricatti equations: where and

  • risk premiums:

  • term premium:

State-Space Reparameterization

State-Space Form

Let be the maturities in the given sample excluding the short rate, and the sample yields be denoted as

Cite

When estimating the Affine Term Structure Model (ATSM), the short rate is often excluded because it can be highly volatile and difficult to predict accurately. The short rate, which is the interest rate for an infinitesimally short period, is influenced by a variety of factors, including central bank policies and market conditions. Instead, ATSMs typically focus on longer-term interest rates, which are more stable and provide a clearer picture of the overall term structure. By excluding the short rate, the model can better capture the dynamics of the yield curve and provide more reliable estimates for pricing and risk management.

Now assume that the short rate has measurement errors with a variance , and the sample yields have measurement errors with a variance .

Then the measurement equations are and the transition equation is

Now following Hamilton and Wu (2012), assume that the short rate, 3-year and 10-year yields are observed without error. Thus we can express these three yields as where and , meaning . Below, we denote

Lee and Kang (2023)

Introducing measurement errors would be more realistic. However, if we include measurement errors, we have to rely on filters that calculate the approximate likelihood, for example, the Kim filter (Kim, 1994) or the Particle filter. In contrast, introducing only measurement errors allows us to calculate the likelihood without approximation. It leads to more accurate calculations of the marginal likelihoods. Since differences between the nine models’ marginal likelihoods can be small numerically, accurate calculation of marginal likelihood is essential for our model comparison. In addition, since the FRB dataset provides yields interpolated by the extension of the DNS model (Gurkaynak, Sack, and Wright, 2007) and our model is the AFNS model, the measurement errors would be very small.

Then, by ordering the yields so that the last maturities be with measurement errors, we have

Thus, the modified state-space is

  • measurement equation:
  • transition equation:

Reparameterization

  • measurement equation: where . Thus which is
  • transition equation: for simplicity, we ignore the physical measure , and denote and . Thus we have

Estimation with Bayesian method

Prior Distribution

The parameters we need to estimate are note that we can obtain , , and .

  • intercept of risk-neutral dynamics:
  • coefficient of risk-neutral dynamics: for matrix ,
  • coefficient of physical dynamics: for matrix ,
  • variance-covariance matrix: and ,
  • measurement error:
  • decay parameter: note that . for , let i.e. and collect such and denote the set as . and, we let

Full-Conditional Distribution

Risk-neutral constant term

From and where and , we have Thus using from the measurement equation, we have As the first 2 rows have no measurement errors, we can directly derive the constant term by

Decay Parameter

Estimation Steps

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Table Of Contents
Macro-Finance ATSM with Bayesian
Unspanned Macro-Finance Model
Macro-Finance Models
Unspanned Model Restrictions
Stylized Facts
Arbitrage-free Nelson Siegel Restrictions
Model Equations
State-Space Reparameterization
State-Space Form
Reparameterization
Estimation with Bayesian method
Prior Distribution
Full-Conditional Distribution
Risk-neutral constant term
Decay Parameter
Estimation Steps